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Appendix 1 Dynamic Polarizability of an Atom

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Appendix 1 Dynamic Polarizability of an Atom Appendix 1 Dynamic Polarizability of an Atom Definition of Dynamic Polarizability The dynamic (dipole) polarizability aoðÞis an important spectroscopic characteris- tic of atoms and nanoobjects describing the response to external electromagnetic disturbance in the case that the disturbing field strength is much less than the atomic << 2 5 4 : 9 = electric field strength E Ea ¼ me e h ffi 5 14 Á 10 V cm, and the electromag- netic wave length is much more than the atom size. From the mathematical point of view dynamic polarizability in the general case is the tensor of the second order aij connecting the dipole moment d induced in the electron core of a particle and the strength of the external electric field E (at the frequency o): X diðÞ¼o aijðÞo EjðÞo : (A.1) j For spherically symmetrical systems the polarizability aij is reduced to a scalar: aijðÞ¼o aoðÞdij; (A.2) where dij is the Kronnecker symbol equal to one if the indices have the same values and to zero if not. Then the Eq. A.1 takes the simple form: dðÞ¼o aoðÞEðÞo : (A.3) The polarizability of atoms defines the dielectric permittivity of a medium eoðÞ according to the Clausius-Mossotti equation: eoðÞÀ1 4 ¼ p n aoðÞ; (A.4) eoðÞþ2 3 a V. Astapenko, Polarization Bremsstrahlung on Atoms, Plasmas, Nanostructures 347 and Solids, Springer Series on Atomic, Optical, and Plasma Physics 72, DOI 10.1007/978-3-642-34082-6, # Springer-Verlag Berlin Heidelberg 2013 348 Appendix 1 Dynamic Polarizability of an Atom where na is the concentration of substance atoms. For simplicity it is assumed in Eq. A.4 that a medium consists of atoms of the same kind. It should be noted that the basis for a number of experimental procedures of determination of the dynamic polarizability aoðÞis its connection with the refractive index of a substance (thatpffiffiffiffiffiffiffiffiffiffi for a transparent nonmagnetic medium is determined by the equation nðÞ¼o eoðÞ). Dynamic polarizability defines the shift of the atomic level energy DEn in an external electric field. In the second order of the perturbation theory for the nonresonant external field E and a spherically symmetric electronic state a corresponding correction to energy looks like 1 DEð2Þ ¼À a ðÞo E2: (A.5) n 2 n The formula (A.5) describes the quadratic Stark effect. In the case that the external field frequency coincides with the eigenfrequency of an atom, the energy shift is found to be linear in electric field intensity – the linear Stark effect. The linear Stark effect is realized also in case of an orbitally degenerate atomic state as it takes place for a hydrogen atom and hydrogen-like ions. Static polarizability, that is, polarizability at the zero frequency aoðÞ¼ 0 defines the level shift in a constant electric field and, besides, the interatomic interaction potential at long distances (the Van der Waals interaction potential). Since static polarizability is a positive value (see below), from the Eq. A.5 it follows that the energy of a nondegenerate atomic state decreases in the presence of a constant electric field. The potential of interaction of a neutral atom with a slow charged particle at long distances is also defined by its static polarizability: að0Þ V ðrÞ¼Àe2 ; (A.6) pol 0 2 r4 where e0 is the particle charge. With the use of Eq. A.6 it is possible to obtain the following expression for the cross-section of elastic collision of a charged particle with an atom in case of applicability of the classical approximation for description of motion of an incident particle with the energy E: rffiffiffiffiffiffiffiffiffi að0Þ sel ðEÞ¼2 p e : (A.7) scat 0 2E It should be noted that the Eq. A.7 follows (accurate to the factor equal to 2) from Eq. A.6 if the effective scattering radius rE is determined with the use of the equation E ¼ VpolðÞrE : (A.8) Appendix 1 Dynamic Polarizability of an Atom 349 Thus the knowledge of dynamic polarizability is very important for description of a whole number of elementary processes. Expression for the Dynamic Polarizability of an Atom Let us calculate the dipole moment of an atom d in the monochromatic field EðtÞ¼2RefgEo expðÞÀi o t that by definition is dðtÞ¼2RefgaðÞo Eo expðÞÀi o t : (A.9) The Fourier component of the dipole moment is given by the expression do ¼ aðÞo Eo: (A.10) In the formulas (A.9) and (A.10) Eo is the complex electric field vector in monochromatic radiation being a Fourier component of EðtÞ. The dipole moment of an atom in the absence of external fields is equal to zero in view of spherical symmetry, so the value of an induced dipole moment can serve as a measure of disturbance of an atom by an external action. The linear dependence dðtÞon electric field intensity (A.9) is true in case of smallness of the field strength E from the standpoint of fulfilment of the inequations E<<Ea. Thus for low enough field strengths the response of an atom to electromagnetic disturbance can be characterized by its polarizability aðÞo . For description of the electromagnetic response of an atom – a quantum system – within the framework of classical physics, it is convenient to use the spectroscopic conformity principle. It can be formulated as follows: an atom in interaction with an electromagnetic field behaves as a set of classical oscillators (transition oscillators) with eigenfrequencies equal to the frequencies of transitions between atomic energy levels. This means that each transition between the atomic states jij and jin is assigned an oscillator with the eigenfrequency ojn and the damping constant djn< <ojn . The contribution of the transition oscillators to the response of an atom to electromagnetic action is proportional to a dimensionless quantity called oscillator strength – fjn , the more is the oscillator strength, the stronger is a corresponding transition. Transitions with the oscillator strength equal to zero are called forbidden transitions. According to the spectroscopic conformity principle, the change of an atomic state is made up of the change of motion of transition oscillators. The inequation E<<Ea means the smallness of perturbation of an atomic electron state as a result of action of the electromagnetic field. Thus it is possible to consider the deviations of the transition oscillators from the equilibrium position under the action of the field EðtÞ small, so for a nth oscillator the equation of motion in the harmonic approximation is true: 350 Appendix 1 Dynamic Polarizability of an Atom e €r þ d r_ þ o2 r ¼ f EðtÞ; (A.11) n n0 n n0 n m n0 where rn is the radius vector corresponding to the deviation of the transition oscillator from the equilibrium position; dn0, on0, fn0 are the damping constant, eigenfrequency and the oscillator strength. For simplicity we consider a one-electron atom in the ground state, the dipole moment of which is equal to d ¼ e r . (In case of a multielectron atom the dipole moment is equal to the sum of dipole moments of atomic electrons.) In view of the conformity principle the induced dipole moment of an atom is made upP of inducedP dipole moments of the oscillators of transitions to the nth state dn: d ¼ dn ¼ e rn. Going in this equation to Fourier components, n n we have X do ¼ e rno; (A.12) n where rno is the Fourier transform of the radius vector of the transition oscillator deviation from the equilibrium position. The expression for this value follows from the equation of motion (A.11): e fn0 r o ¼ Eo: (A.13) n o2 o2 o m n0 À À i dn0 Substituting the formula (A.13) in the Eq. A.12 and using the definition of polarizability (A.10), we find for it the following expression: e2 X f aðÞ¼o n0 : (A.14) o2 o2 o m n n0 À À i dn0 Hence it follows that the dynamic polarizability of an atom represents, generally speaking, a complex value with a dimensionality of volume. The imaginary part of polarizability is proportional to the damping constants of the transition oscillators. The sum on the right of the Eq. A.14 includes both summation over the discrete energy spectrum and integration with respect to the continuous energy spectrum. The imaginary part of polarizability is responsible for absorption of radiation, and the real part defines the refraction of an electromagnetic wave in a medium. The expression (A.14) describes not only a one-electron atom, but also a multielectron atom. The multielectron nature of an atom is taken into account by the fact that in definition of the oscillator strength the dipole moment of an atom is equal to the sum of dipole moments of its electrons. From the Eq. A.14 several important limiting cases can be obtained. For example, if the frequency of the external field is equal to zero, the formula (A.14) gives the expression for the static polarizability of an atom: Appendix 1 Dynamic Polarizability of an Atom 351 e2 X f a aðÞ¼o ¼ 0 n0 : (A.15) 0 o2 m n n0 Hence it is seen that static polarizability is a real and positive value. It has a large numerical value if in the atomic spectrum there are transitions with high oscillator strength and low eigenfrequency as it is, for example, for alkaline-earth atoms. In the opposite high-frequency limit, when h o>>IP ( IP is the ionization potential of atom) and the eigenfrequencies in the denominators of Eq. A.14 can be neglected, from the formula (A.14) in view of the golden rule of sums, according to which the sum of oscillator strengths is equal to the number of electrons in an atom Na, we obtain e2 N a ðÞ¼Ào a : (A.16) 1 m o2 Hence it is seen that the high-frequency polarizability of an atom is a real and negative value that decreases quadratically with growing frequency of the external field.
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